You've asked only about the show, but this is never explicitly stated on in the show. As far as I can tell the shows information on this comes from the books, as such I've answered from a books perspective.
A Song of Ice and Fire
The Wall was built by Brandon the Builder, or so the legends say.
"I could tell you the story about Brandon the Builder," Old Nan said. "That was always your favorite."
Thousands and thousands of years ago, Brandon the Builder had raised Winterfell, and some said the Wall. Bran knew the story, but it had never been his favorite. Maybe one of the other Brandons had liked that story.
A Game of Thrones, Bran IV
The Wall was often said to stand seven hundred feet high, but Jarl had found a place where it was both higher and lower. Before them, the ice rose sheer from out of the trees like some immense cliff, crowned by wind-carved battlements that loomed at least eight hundred feet high, perhaps nine hundred in spots. But that was deceptive, Jon realized as they drew closer. Brandon the Builder had laid his huge foundation blocks along the heights wherever feasible, and hereabouts the hills rose wild and rugged.
A Storm of Swords, Jon IV
The suggestion outraged some of the others. "Do you want the king to wipe our arses for us too?" said Cotter Pyke angrily. "The choice of a Lord Commander belongs to the Sworn Brothers, and to them alone," insisted Ser Denys Mallister. "If they choose wisely they won't be choosing me," moaned Dolorous Edd. Maester Aemon, calm as always, said, "Your Grace, the Night's Watch has been choosing its own leader since Brandon the Builder raised the Wall. Through Jeor Mormont we have had nine hundred and ninety-seven Lords Commander in unbroken succession, each chosen by the men he would lead, a tradition many thousands of years old."
A Storm of Swords, Samwell V
It is said that Brandon seeked the aid of the Children of the Forest to raise the Wall.
Their song and music was said to be as beautiful as they were, but what they sang of is not remembered save in small fragments handed down from ancient days. Maester Childer's Winter's Kings, or the Legends and Lineages of the Starks of Winterfell contains a part of a ballad alleged to tell of the time Brandon the Builder sought the aid of the children while raising the Wall. He was taken to a secret place to meet with them, but could not at first understand their speech, which was described as sounding like the song of stones in a brook, or the wind through leaves, or the rain upon the water. The manner in which Brandon learned to comprehend the speech of the children is a tale in itself, and not worth repeating here. But it seems clear that their speech originated, or drew inspiration from, the sounds they heard every day.
The World of Ice and Fire, Ancient History: The Dawn Age
It was built to keep out the Others and infused with more than just Ice and stone, it contained magic.
"The Wall. The Wall is more than just ice and stone, he said. There are spells woven into it . . . old ones, and strong. He cannot pass beyond the Wall."
A Storm of Swords, Bran IV
"You are wrong. I have dreamed of your Wall, Jon Snow. Great was the lore that raised it, and great the spells locked beneath its ice. We walk beneath one of the hinges of the world." Melisandre gazed up at it, her breath a warm moist cloud in the air. "This is my place as it is yours, and soon enough you may have grave need of me. Do not refuse my friendship, Jon. I have seen you in the storm, hard-pressed, with enemies on every side. You have so many enemies. Shall I tell you their names?"
A Dance with Dragons, Jon I
GRRM has also commented on the fact that Brandon the Builder may just be a legend but leaves it open for interpretation still.
If time is permiting would you mind giving a brief description on how the wall was constructed?
Much of those details are lost in the mists of time and legend. No one can even say for certain if Brandon the Builder ever lived. He is as remote from the time of the novels as Noah and Gilgamesh are from our own time.
But one thing I will say, for what it's worth -- more than ice went into the raising of the Wall. Remember, these are =fantasy= novels.
So Spake Martin, The Wall